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c Indian Academy of Sciences PRAMANA — journal of physics Vol. 59, No. 2 August 2002 pp. 175–179 Spin squeezing and quantum correlations K S MALLESH1 , SWARNAMALA SIRSI2 , MAHMOUD A A SBAIH1 , P N DEEPAK1 and G RAMACHANDRAN3 1 Department of Studies in Physics, University of Mysore, Mysore 570 006, India Department of Physics, Yuvaraja’s College, University of Mysore, Mysore 570 005, India 3 Indian Institute of Astrophysics, Bangalore 560 034, India 2 Abstract. We discuss the notion of spin squeezing considering two mutually exclusive classes of spin-s states, namely, oriented and non-oriented states. Our analysis shows that the oriented states are not squeezed while non-oriented states exhibit squeezing. We also present a new scheme for construction of spin-s states using 2s spinors oriented along different axes. Taking the case of s = 1, we show that the ‘non-oriented’ nature and hence squeezing arise from the intrinsic quantum correlations that exist among the spinors in the coupled state. Keywords. Squeezing; spin; quantum correlation. PACS Nos 03.65.-W; 03.65.Ta 1. Introduction It is well-known that the state of a harmonic oscillator is said to be squeezed if the variance ∆x2 or ∆p2 is less than 12 which is the minimum uncertainty limit. Although squeezing is thus unambiguously defined in the case of bosonic systems [1] its definition in the context of spin needs careful consideration. A comparison of the uncertainty relations satisfied by the components of the spin operator ~S, ∆Sx2∆Sy2 hSzi2 4 ; x; y; z cyclic; (1) with ∆x2 ∆p2 1 ; 4 (2) would naturally suggest that a spin state could be regarded as squeezed if ∆S x2 or ∆Sy2 is smaller than jhSz ij=2, where the expectation value and the variances are calculated in some arbitrary coordinate system. Indeed this has been used as the squeezing criterion in the literature [2]. Such a definition does not take into consideration the existence of quantum correlations and is coordinate dependent. In an attempt to arrive at a proper criterion for squeezing, Kitagawa and Ueda [2] have considered a model in which a spin-s 175 K S Mallesh et al state is visualized as being built out of 2s elementary spin- 12 states. A coherent spin-s state (CSS) jθ ; φ i can then be thought of as having no quantum correlations as the constituent 2s elementary spins point in the same direction n̂(θ ; φ ); which is the mean spin direction. 2. State classification and squeezing In order to discuss squeezing, we begin with the squeezing condition itself. Referring to [2,4] we adopt the following definition: A spin-s state is squeezed in the spin component normal to the mean spin direction n̂ if ∆ ~S:n̂? 2 < jhS n̂ij ~: 2 ; hSi hSi hSi ~ n̂ = q ~ : ~ ; n̂:n̂? = 0: (3) It is easy to see that the familiar angular momentum states jsmi n̂ are not squeezed. But one can however consider superpositions of the states jsmi k̂ of the form jψ i = ∑ Cm jsmik̂ (4) ; m and investigate if these exhibit squeezing or not. For this purpose, we classify such states into two mutually exclusive classes, namely, the oriented and non-oriented states, which together exhaust all pure states in the 2s + 1 dimensional spin space of the system. An oriented spin state by definition is a state jψ i of the form jψ i = jsm0 ik̂0 = ∑ Dsmm0 (αβ γ )jsmik̂ (5) : m Here Ds denote the standard rotation matrices and α ; β ; γ are the Euler angles taking î jˆk̂ to î0 jˆ0 k̂0 . If we now calculate the variance perpendicular to the mean spin direction, it indeed turns out to be exactly equal to ∆ ~S:n̂? 2 = 1 s (s + 1) 2 m02 (6) ; which is never less than 12 jh~S:n̂ij. Thus no oriented pure state is a squeezed state. Any normalized spin-s state jψ i of the form (4) is, in general, specified by 4s real independent parameters. The oriented states described above are specified at the most by the three independent Euler angles α ; β and γ . Since 4s > 3, for s 1, there exist states which are not oriented. In other words, there exist states which can not be identified as eigen states of S 2 and Sz with respect to any choice of the axis of quantization. We refer to such states as non-oriented. While an oriented state is characterized by a single direction, viz., the axis of quantization (specified by two real variables θ ; φ ) in the physical space, a non-oriented state could be characterized by more than one direction. In order to see whether squeezing exists for a non-oriented state we now start with an arbitrary state jψ i and first determine its mean spin direction ẑ 0 . The most general spin-1 state that possesses a non-zero mean spin value h~Si, can be written in the form jψ i = cos δ j1 1iẑ ; 176 0 + sin δ j1 ; 1iẑ 0 ; 0<δ < π; Pramana – J. Phys., Vol. 59, No. 2, August 2002 (7) Spin squeezing and quantum correlations where j1; m0 iẑ are the angular momentum states specified with respect to zˆ0 . This state is 0 obviously non-oriented for all values of δ other than 0; π4 ; π2 ; 34π ; π . For such a state referred to the frame x 0 y0 z0 , the squeezing conditions for S x0 and Sy0 are respectively given by 1 + sin2δ < sin2δ < j cos 2δ j (8) j cos 2δ j (9) and 1 : These conditions are indeed separately valid for the entire range except for δ π π 3π 4 ; 2 ; 4 ; π , which implies that a non-oriented state jψ i is indeed a squeezed state. = 0, 3. Quantum correlations Having thus identified the squeezed states in the spin-1 case, it is of interest to analyse in quantitative terms if squeezing in spin systems arises from the existence of quantum correlations. This can be done by employing the model in which a spin-s state is constructed using 2s spin- 12 states. Majorana’s geometric realization [5] of a spin-s state as a constellation of 2s points on a sphere leads to Schwinger’s idea [6] of realising jsmi states in the form a†+ jsmi = s+m a† ((s + m)! (s s m 1 m)!) 2 j00i (10) ; where a†+ ; a† are the creation operators for the spin ‘up’ and spin ‘down’ states, respectively. It must be noted here that spin ‘up’ and spin ‘down’ states as well as jsmi states are all referred to the same axis of quantization. At this point, we would like to generalize this realization by taking the 2s ‘up’ spinors u(θl ; φl ); l = 1; :::; 2s, where the kth spinor is specified with respect to an axis of quantization Q̂k (θk φk ) in the physical space. Coupling 2s spin- 12 states in this way leads to a spin-s state in the form (4), where the coefficients C m are given by Cm = Ns dm ; Ns 1 s = ∑ jdm j2 !1=2 (11) m= s and dm = ∑ m1 ;:::;m2s 1 D m 1 2 1 12 C( 12 12 1; m1 m2 µ1 )C(1 12 32 ; µ1 m3 µ2 ) C(s (φ1 θ1 0) Dm 1 2 1 2s 2 (φ2s θ2s 0): 11 2 2 s; µ2s 2 m2s m) (12) Thus our construction of a spin-s state jψ i is done using 2s spin- 12 states which are specified with respect to 2s different directions, Q̂1 ; Q̂2 ; : : : ; Q̂2s in general. In particular, if Q̂1 = Q̂2 = = Q̂2s , then our construction specializes to the realization suggested Pramana – J. Phys., Vol. 59, No. 2, August 2002 177 K S Mallesh et al by Schwinger and employed in refs [2–4]. Indeed, in this particular case, the spin state realized is nothing but an oriented state jsmi. The significance of our construction lies in the fact that if Q̂l 6= Q̂m for at least two quantization directions, the state realized is a non-oriented state of spin-s. Considering in particular the simplest case of s = 1, we note that such a construction can be carried out using two spinors specified with respect to Q̂1 (θ1 φ1 ) and Q̂2 (θ2 φ2 ) so that the spin-1 state jψ i = N1 ∑ Dm1 21 2(φ1 θ1 0)Dm1 2 (φ2 θ2 0)C( 12 12 1; m1 m2 m)j( 12 12 )1mi = m1 ;m 1 = 1 22 = (13) in the lab frame î jˆk̂ is non-oriented if Q̂1 6= Q̂2 . The mean spin direction ẑ 0 for such a state happens to be along the bisector of the two directions Q̂1 and Q̂2 . The squeezing condition for S x0 now takes the form cos2 θ < j cos θ j (14) which is satisfied for all θ except when θ = 0, π =2, π . The absence of squeezing for θ = 0; π =2; π is obvious as the two axes then merge together giving an oriented state. Thus in all other cases the state jψ i is squeezed in the spin component S x0 and is non-oriented by construction. We now establish explicitly for s = 1, the connection between squeezing and the spin– spin correlations that exist between the component spinors. Any spin-1 state constructed using the two spinors is said to possess spin correlations if the matrix C 12 defined through its elements 12 Cµν = ∆(S1 µ S2ν ) = hS1µ S2ν i hS1µ ihS2ν i (15) is non-zero. Here S 1µ and S2ν are the spin components associated with the two spinors and the angular brackets denote the expectation values with respect to the coupled state. For the state jψ i in (7), the correlation matrix is diagonal in the frame x 0 y0 z0 with the ‘diagonal’ or the ‘eigen’ correlation elements given by Cx12x = 0 0 sin2 θ 4(1 + cos2 θ ) = Cy12y ; 0 0 Cz12z = 0 0 sin2 θ 2(1 + cos2 θ ) 2 : (16) A glance at these expressions shows that when θ = 0; π =2; π , the values of the correlations are either 0 or 1=4. On the other hand for all other values of θ , the eigen correlations satisfy 0 < jCii12 j < 1=4; i = x0 ; y0 ; z0 : (17) In other words, all non-oriented spin-1 states have the eigen correlations restricted to the above range. One can also see that the trace of the correlation matrix is Tr(C 12 )= sin2 θ 2(1 + cos2 θ ) 2 : This being invariant under rotations of the coordinate frames, satisfies the condition 178 Pramana – J. Phys., Vol. 59, No. 2, August 2002 (18) Spin squeezing and quantum correlations 0 Tr(C 12 ) 1=4: (19) Indeed if a given coupled state has a correlation matrix that satisfies this condition, the state is squeezed. We can find the value of θ through cos θ = " 1 p 2 Tr(C12 ) p 1 + 2 Tr(C12 ) #1=2 ; (20) which identifies the structure of the state in terms of the two spinors. The four values of θ that satisfy the above equation correspond to the directions Q̂1 and Q̂2 . Thus we conclude that the trace condition (19) on the correlation matrix is the necessary and sufficient condition for a spin-1 state to be squeezed. 4. Conclusions We have classified spin states into two mutually exclusive classes, namely, oriented and non-oriented states, and studied their squeezing properties. It is clear from our analysis that squeezing is exhibited only by non-oriented states. Considering in particular the nonoriented states of a spin-1 system, we have shown that they exhibit squeezing. This has been illustrated in two different ways: first by looking at the non-oriented nature of the spin-1 state itself, and secondly, by introducing a new form of coupling in which two spin1 2 states add up to give the required spin-1 non-oriented state. Our construction gives a quantitative description of the existence of quantum correlations as well as an indication as to how they lead to non-oriented nature and hence to the squeezing behavior. This intimate relationship between squeezing and ‘non-oriented’ nature indeed suggests a way to prepare a squeezed state. The non-oriented states are potential candidates for observing squeezing experimentally. A recent study by Ramachandran and Deepak [7] reveals that the collision of a spin- 12 beam with a spin- 12 target, both oriented in different directions, leads to a combined spin state which is non-oriented. Acknowledgements The authors (GR and PND) acknowledge with thanks the financial support of CSIR, India. References [1] [2] [3] [4] H J Kimble and D F Walls (eds), J. Opt. Soc. B4, 1450 (1987) M Kitagawa and M Ueda, Phys. Rev. A47, 5138 (1993) D J Wineland et al, Phys. Rev. A46, 6797 (1992) R R Puri, Pramana – J. Phys. 48, 787 (1997) G S Agarwal and R R Puri, Phys. Rev. A49, 4968 (1994) [5] E Majorana, Nuovo Cimento 9, 43 (1932) [6] J Schwinger, in Quantum theory in angular momentum edited by L C Biedenharn and H Van Dam (Academic Press, NY, 1965) [7] G Ramachandran and P N Deepak, Proc. DAE Symp. on Nucl. Phys. edited by V M Datar and A B Santra, B40, 300 (1997) Pramana – J. Phys., Vol. 59, No. 2, August 2002 179